Competing length scales and 2D versus 3D dimensionality in relatively thick superconducting NbN films

Magneto-transport characteristics of 2D and 3D superconducting layers, in particular, temperature and angular dependences of the upper critical field Hc2, are usually considered to be fundamentally different. In the work, using non-local resistance measurements at temperatures near the normal-to-superconducting transition, we probed an effective dimensionality of nm-thick NbN films. It was found that in relatively thick NbN layers, the thicknesses of which varied from 50 to 100 nm, the temperature effect on Hc2 certainly pointed to the three-dimensionality of the samples, while the angular dependence of Hc2 revealed behavior typical for 2D samples. The seeming contradiction is explained by an intriguing interplay of three length scales in the dimensionally confined superconducting films: the thickness, the Ginzburg–Landau coherence length, and the magnetic-field penetration depth. Our results provide new insights into the physics of superconducting films with an extremely large ratio of the London penetration depth to the Ginzburg–Landau coherence length exhibiting simultaneously 3D isotropic superconducting properties and the 2D transport regime.


NbN film fabrication and characterization
The pulsed laser deposition of NbN films was carried out in an ultrahigh vacuum chamber using an excimer KrF laser with the 248 nm wavelength, the pulse duration of 35 ns, and the laser fluency of 4.94 Jcm -2 .The NbN thin films were deposited on c-cut Al2O3 substrates, ultrasonically cleaned in acetone, isopropanol and deionized water, from a 2-inch Nb target (99.9%) in the N2 + 1% H2 reactive atmosphere at the gas flow of 80 sccm and the pressure of 9.3 Pa.The substrate temperature was kept constant at 600 o C.
After deposition, the NbN thin films were characterized in detail by several analytical techniques including X-ray diffraction, X-ray reflectivity, scanning electron microscopy, energydispersive X-ray spectroscopy and atomic force microscopy.A complete set of characterization results for the most studied 50-nm thick layers can be found in Refs.S1 and S2.Below we compare θ/2θ XRD patterns for three NbN films with thicknesses of 10, 50, and 100 nm whose magnetotransport properties are analyzed in the work.In all three samples, we see a very strong (111) preferential orientation of fcc-NbN on c-cut Al2O3 substrate.The XRD spectra also show sideoscillations, especially clearly visible on the 10 nm thick sample due to perfect lattice coherency throughout its thickness.The period of oscillations corresponds to the thickness of each film indicating a high degree of out-of-plane crystallinity.In general, the crystal structure appears to be identical, although some subtle details require further study.Our approach to analyzing the superconducting heterogeneity in thin current-conducting films goes back to the well-known modification of the four-probe method for measuring the in-plane sheet resistance which was proposed in 1958 by van der Pauw [S3].Solving the general problem of potential distribution in thin conductive layers of any shape and placing four probes at the film edges made it possible to carry out measurements that give a weighted average of local resistances of the sample, whereas the traditional linear layout of the four contacts provides knowledge of resistivity only in the probing direction [S4].In the van der Pauw arrangement, a current is flowing along one edge of the sample and the voltage across the opposite edge is measured, see Fig. S5 below.From these two values, the four-probe resistance can be found using conventional Ohm's law.
However, for correct measurement, certain conditions must be met.The thickness d of the film, which should not have holes and/or islands of highly conductive material, has to be uniform and small in relation to the distance between the probes.And most importantly, the samples must be homogeneous and isotropic.Our control measurements of a local normal-state resistance using the standard in-line contact arrangement showed that the samples are quite homogeneous from the viewpoint of the normal state, but there is a noticeable spread in the critical temperatures and transition widths in the superconducting state.The latter circumstance has already been noted in the literature, specifically, for niobium nitride films [S5].
The authors of the work [S4] drew attention to the fact that in rather inhomogeneous samples the van der Pauw method [S3] can lead to non-physical results and, to confirm this, proposed a simplified four-resistor model.In the main text of the paper, we follow the model with a square thin layer conditionally divided into four resistive regions [S4] and consider an equivalent electrical circuit with a current source and four resistors Ri (i = 1,2,3,4).

Figure S3 .
Figure S3.Four-probe resistance-vs-temperature traces for a representative 50-nm thick NbN film

Figure S5 .
Figure S5.Equivalent four-resistor circuit for van der Pauw resistive measurements of a square superconducting layer conditionally divided into four resistive regions